From a shariah perspective, it is not prohibited (as long as one is not intentionally doing it to insult Allah or Ahl al-Bayt).
However, it is better not to do it in the first 10 days of Muharram, or on Ashura, if there is a choice. (For instance, you have not been evicted.)
From a hadith perspective, those are the times most emphasised for remembering Imam Husayn (A), but it is customary these days to observe all of Muharram and Safar.
So in general, it is better not to move during Muhrram, if it is optional and can be postponed, but sometimes there is no choice, or there is a strong reason to move. However, this is a matter of personal piety and observance, not shariah.
One thing to consider is that some people may be concerned that their new home lacks baraka if they move during this time. Then, if something goes wrong (and something always goes wrong in life!), they will assume this is because they moved during Muharram, and may not feel settled in their home. Even though this is a psychological reason, it may be better to avoid moving during this time for that reason.
Also, in some communities, people will be criticized if they do it during this time, and one has to decide whether or not to take into consideration the views of others.
In any case, it is your decision. Wishing you the best however it goes!
